Narrow Results

Since the validity of the Navier-Stokes equations is well established, any fluid dynamic phenomenon could be calculated if methods for solving them correctly are obtained. A fair portion of this dream seems to have come true through the remarkable development of supercomputers and solution algorithms that have made the simulation of high-Reynolds-number flows possible. For understanding the underlying flow mechanism, means of properly visualizing the computed flow field are needed and have been developed. On the whole, computer simulation is becoming the most effective tool for the study of fluid dynamics.

We extend the analysis of Ref. 16 showing statistically significant log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number (≈ 2500) of three-dimensional fully developed turbulence. First, we develop a simple variant of the canonical averaging method using a rephasing scheme between different samples based on pairwise correlations that confirms Zhou and Sornette's previous results. The second analysis uses a simpler local spectral approach and then performs averages over many local spectra. This yields stronger evidence of the existence of underlying log-periodic undulations, with the detection of more than 20 harmonics of a fundamental logarithmic frequency f = 1.434 ± 0.007 corresponding to the preferred scaling ratio γ = 2.008 ± 0.006.

A detailed analysis on the characteristics of transitional turbulent flow over a bell-shape stenosis for a physiological pulsatile flow is presented. The comparison of the numerical solutions to three types of pulsatile flows, including a physiological flow, an equivalent pulsatile flow and a simple pulsatile flow, are made in this work. Then the effects of the Reynolds number, Womersley number and constriction ratio of stenosis on the pulsatile turbulent flow fields for the physiological flow are considered. The comparison of the three pulsatile flows shows that the flow characteristics cannot be properly estimated if an equivalent or simple pulsatile inflow is used instead of actual physiological one in the study of the pulsatile flows through arterial stenosis. The equivalent or simple pulsatile inflow can lead to higher disturbance intensity in the vicinity of the stenosis than the physiological inflow. For a physiological flow, the recirculation zones with high disturbance intensity occur mainly in the distal of the stenosis. The larger Reynolds number and severer constriction ratio may result in more complex flow field and cause some important flow variables to increase dramatically near stenosis. The higher Womersley number leads to a larger phase lag between the imposed flow rate changes and the final converged flow field in one cycle. The turbulence intensity decreases with the increase of Womersley number for the same Reynolds number.

A numerical experiment is presented that is taken from the recent literature. It has been devised as a benchmark case to test the quality of boundary conditions in numerical solvers for computational fluid dynamics. In this experiment, a two-dimensional system of two counter-rotating vortexes is brought into collision with a no-slip wall and rebounds from it. In the present paper, the benchmark is run with a lattice-Boltzmann numerical solver. Astonishingly accurate results are obtained with a straightforward boundary condition known under the name of bounce-back. This sample problem is also used to discuss techniques for the setup of an initial condition in the lattice Boltzmann method.

Based on a formal analogy between space-time quantum fluctuations and classical Kolmogorov fluid turbulence, we suggest that the dynamic growth of the Universe from Planckian to macroscopic scales should be characterized by the presence of a fluctuating volume-flux (FVF) invariant. The existence of such an invariant could be tested in numerical simulations of quantum gravity, and may also stimulate the development of a new class of hierarchical models of quantum foam, similar to those currently employed in modern phenomenological research on fluid turbulence. The use of such models shows that the simple analogy with Kolmogorov turbulence is not compatible with a fine-scale fractal structure of quantum space-time. Hence, should such theories prove correct, they would imply that the scaling properties of quantum fluctuations of space-time are subtler than those described by the simple Kolmogorov analogy.

The motivation for this work was a simple experiment [P. M. C. de Oliveira, S. Moss de Oliveira, F. A. Pereira and J. C. Sartorelli, preprint (2010), arXiv:1005.4086], where a little polystyrene ball is released falling in air. The interesting observation is a speed breaking. After an initial nearly linear time-dependence, the ball speed reaches a maximum value. After this, the speed finally decreases until its final, limit value. The provided explanation is related to the so-called von Kármán street of vortices successively formed behind the falling ball. After completely formed, the whole street extends for some hundred diameters. However, before a certain transient time needed to reach this steady-state, the street is shorter and the drag force is relatively reduced. Thus, at the beginning of the fall, a small and light ball may reach a speed superior to the sustainable steady-state value.

Besides the real experiment, the numerical simulation of a related theoretical problem is also performed. A cylinder (instead of a 3D ball, thus reducing the effective dimension to 2) is positioned at rest inside a wind tunnel initially switched off. Suddenly, at t = 0 it is switched on with a constant and uniform wind velocity far from the cylinder and perpendicular to it. This is the first boundary condition. The second is the cylinder surface, where the wind velocity is null. In between these two boundaries, the velocity field is determined by solving the Navier–Stokes equation, as a function of time. For that, the initial condition is taken as the known Stokes laminar limit V → 0, since initially the tunnel is switched off. The numerical method adopted in this task is the object of the current text.

Recently, it was shown that energy conserving (EC) lattice Boltzmann (LB) model is more accurate than athermal LB model for high-resolution simulations of athermal flows. However, in the sub-grid (SG) domain, the behavior is found to be opposite. In this work, we show that via multi-relaxation model, it is possible to preserve the accuracy of the EC LB for both SG and direct numerical simulation (DNS) models. We show that by introducing a nonunit Prandtl number, under-resolved simulations can also be performed quite efficiently, a property which we attribute to the enhanced sound-relaxation.

Large-scale molecular dynamics (MD) simulations of freely decaying turbulence in three-dimensional space are reported. Fluid components are defined from the microscopic states by eliminating thermal components from the coarse-grained fields. The energy spectrum of the fluid components is observed to scale reasonably well according to Kolmogorov scaling determined from the energy dissipation rate and the viscosity of the fluid, even though the Kolmogorov length is of the order of the molecular scale.

We calculate the power spectral density and velocity correlations for a turbulent flow of a fluid inside a duct. Turbulence is induced by obstructions placed near the entrance of the flow. The power spectral density is obtained for several points at cross-sections along the duct axis, and an analysis is made on the way the spectra changes according to the distance to the obstruction. We show that the differences on the power spectral density are important in the lower frequency range, while in the higher frequency range, the spectra are very similar to each other. Our results suggest the use of the changes on the low frequency power spectral density to identify the occurrence of obstructions in pipelines. Our results show some frequency regions where the power spectral density behaves according to the Kolmogorov hypothesis. At the same time, the calculation of the power spectral densities at increasing distances from the obstructions indicates an energy cascade where the spectra evolves in frequency space by spreading the frequency amplitude.

This paper aims to detect memory loss of the symmetry of blockades in ducts and how far the information on the asymmetry of the obstacles travels in the turbulent flow from computational simulations with OpenFOAM. From a practical point of view, it seeks alternatives to detect the formation of obstructions in pipelines. The numerical solutions of the Navier–Stokes equations were obtained through the solver PisoFOAM of the OpenFOAM library, using the large Eddy simulation (LES) for the turbulent model. Obstructions were placed near the duct inlet and, keeping the blockade ratio fixed, five combinations for the obstacles sizes were adopted. The results show that the information about the symmetry is preserved for a larger distance near the ducts wall than in mid-channel. For an inlet velocity of 5m/s near the walls the memory is kept up to distance 40 times the duct width, while in mid-channel this distance is reduced almost by half. The maximum distance in which the symmetry breaking memory is preserved shows sensitivity to Reynolds number variations in regions near the duct walls, while in the mid channel that variations do not cause relevant effects to the velocity distribution.

The population growth in big urban centers generates the necessity for tall buildings. This phenomenon happens also in tourist regions where it is necessary to host many people. However, locations with high buildings interfere with the flow of the wind and can affect the comfort and safety of pedestrians at street level. Tall buildings barrier reduces the natural ventilation in regions far from the beach. This work presents the results concerning the effects created by tall buildings on Mucuripe beach, Fortaleza, Brazil. We performed numerical simulations to verify the wind interference with buildings in an area of $1.6\times 10^5$m², using the OpenFOAM toolbox, to solve the Reynolds Averaged Navier–Stokes (RANS) equations with the $k$–$ϵ$ turbulence model. The results showed how the obstacles alter the airflow. From them, one can identify the regions with reduced safety and pedestrian comfort, and also the weak wind zone created by the downstream of the constructions for the different wind directions that are locally observed.

This paper focuses on large eddy simulation of Rayleigh–Taylor instability (RTI) based on helicity model (HM) which is derived from our previous research. Based on the RTI problem, we obtain from a priori test and theoretical analysis that the suggested model can automatically distinguish laminar flow and turbulence. Comparing the numerical simulation results with direct numerical simulation (DNS) and traditional model in RTI problem, we could find that the suggested model overcomes some defects of the traditional model, such as the traditional model cannot predict transition accurately and the suggested model can predict the turbulence mixing width and mass fraction precisely. It provides a beneficial tool for the research of RTI.

We indicate a limit of the post-Newtonian gravity equations with incompressible-fluid matter source in which Newtonian gravity is approximately decoupled from relativistic gravitomagnetic effects. In this gravito-magneto-hydrodynamic limit, we perform fully developed turbulence calculations. We demonstrate that gravitomagnetic effects reduce the vortical complexity and nonlinearity of turbulence, even leading to its extinction within large volumes, and generate departures from Kolmogorov turbulence scalings, that are explained via a combination of dimensional and exact analysis arguments.

In this paper, a new configuration of a horizontal axis wind turbine with three blades for domestic electricity production on farms was numerically simulated. Three configurations were tested numerically: a standard single-blade wind turbine (STWT), a single wind turbine (SWT), and a tandem three-blade wind turbine (TWT) for rural use. For the tandem wind turbine, the effects of pitch angle and the distance between the blades were also investigated. The Gambit and Fluent 19.2 codes were, respectively, used to generate different meshes and determine various parameters. The resolution of the averaged Navier–Stokes equations (RANS) using a finite volume method was conducted. The k–$\epsilon$ realizable two-equation model was chosen for turbulence calculation. The obtained results showed that the maximum power coefficient (Cp) was 0.444 for one of the tandem configurations (TWT), compared to the conventional blade (STWT). The maximum Cp obtained in this study is slightly greater than the experimental results found in the literature. It has also been observed that the flow on these new turbines exhibits a complex phenomenon on both surfaces.

Using a classic example proposed by G. I. Taylor, we reconsider through the use of computer algebra, the mathematical analysis of a fundamental process in turbulent flow, namely: How do large scale eddies evolve into smaller scale ones to the point where they are effectively absorbed by viscosity? The explicit symbolic series solution of this problem, even for cleverly chosen special cases, requires daunting algebra, and so numerical methods have become quite popular. Yet an algebraic approach can provide substantial insight, especially if it can be pursued with modest human effort.

The specific example we use dates to 1937 when Taylor and Green⁸ first published a method for explicitly computing successive approximations to formulas describing the three-dimensional evolution over time of what is now called a Taylor–Green vortex.

With the aid of a computer algebra system, we have duplicated Taylor and Green's efforts and obtained more detailed time-series results. We have extended their approximation of the energy dissipation from order 5 in time to order 14, including the variation with viscosity.

Rather than attempting additional interpretation of results for fluid flow (for which, see papers by Brachet et al.,^2,3 we examine the promise of computer algebra in pursuing such problems in fluid mechanics.

We derive a formula for the entropy of two-dimensional incompressible inviscid flow, by determining the volume of the space of vorticity distributions with fixed values for the moments Q_k = ∫ ω(x)^kd²x. This space is approximated by a sequence of spaces of finite volume, by using a regularization of the system that is geometrically natural and connected with the theory of random matrices. By taking the limit we get a simple formula for the entropy of a vortex field. We predict vorticity distributions of maximum entropy with given mean vorticity and enstrophy; we also predict the cylindrically symmetric vortex field with maximum entropy. This could be an approximate description of a hurricane.

Large-scale spatial variations of the guide magnetic field of interplanetary and interstellar plasmas give rise to the adiabatic focusing term in the Fokker–Planck transport equation of cosmic rays. As a consequence of the adiabatic focusing term, the diffusion approximation to cosmic ray transport in the weak focusing limit gives rise to first-order Fermi acceleration of energetic particles if the product HL of the cross helicity state of Alfvenic turbulence H and the focusing length L is negative. The basic physical mechanisms for this new acceleration process are clarified and the astrophysical conditions for efficient acceleration are investigated. It is shown that in the interstellar medium this mechanism preferentially accelerates cosmic ray hadrons over 10 orders of magnitude in momentum. Due to heavy Coulomb and ionization losses at low momenta, injection or preacceleration of particles above the threshold momentum p_c≃0.17Z^2/3GeV/c is required.

We propose a string theory of turbulence that explains the Kolmogorov scaling in 3+1 dimensions and the Kraichnan and Kolmogorov scalings in 2+1 dimensions. This string theory of turbulence should be understood in light of the AdS/CFT dictionary. Our argument is crucially based on the use of Migdal's loop variables and the self-consistent solutions of Migdal's loop equations for turbulence. In particular, there is an area law for turbulence in 2+1 dimensions related to the Kraichnan scaling.

The turbulence in incompressible fluid is represented as a field theory in 3 dimensions. There is no time involved, so this is intended to describe stationary limit of the Hopf functional. The basic fields are Clebsch variables defined modulo gauge transformations (symplectomorphisms). Explicit formulas for gauge invariant Clebsch measure in space of generalized Beltrami flow compatible with steady energy flow are presented. We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the “instantons and intermittency” program we started back in the 1990s.¹ These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier–Stokes equation smears this delta function to the Gaussian with width $h\propto {\nu }^{3/5}$ at $\nu \to 0$ with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation ${\mathrm{\Gamma }}_C$ around fixed loop $C$ in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations² including pre-exponential factor $1/\sqrt{\left|\mathrm{\Gamma }\right|}$. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions $\zeta (n)$ can be fitted at small $n$ to the parabola, coming from the Liouville theory with two parameters $\alpha$, $Q$. At large $n$ the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of $\zeta (\infty )$ in agreement with DNS.

We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as $\mathrm{\Delta }{\overrightarrow{v}}_t=\overrightarrow{\nabla }\mathrm{\Gamma }$. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement $\delta r_{\perp }$ of the vortex sheet as a Hamiltonian coordinate and $\mathrm{\Gamma }$ as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as ${\mathrm{\text{Re}}}^{-\frac{1}{3}}$ with Reynolds number $\mathrm{\text{Re}}\sim {\nu }^{-\frac{6}{5}}$ in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called $c=1$ critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here.